Over the years a number of people have written books and articles in order to popularise mathematics --- from high school mathematics to "higher" mathematics.
The kind of "popular mathematics" that interests me is more in the sense of "popular mechanics" or "popular electronics". Something that gives the reader new things to do and helps build tools to explore further. Just as an essay on Newton or Hook is not an article on popular mechanics, and an essay on Marconi or Baird is not popular electronics, so a book like that by E. T. Bell on the lives of famous (dare I say "great") mathematicians does not constitute "popular mathematics". (For example, this essay is not (yet!) popular mathematics.)
Another aspect of "popular mathematics" is that mathematics should be its own narrative. An article on "building your own rubber-band powered airplane" that spent a lot of time discussing the joys and benefits of flying, would not interest me. In the same sense, an article that extols the beautiful applications of what one is about to study is wasting time in getting to the point --- if it is beautiful it will speak for itself.
There is certainly some interest in how the ideas (might have) come about. A pure recipe, which says, do this calculation followed by this substitution to have this formula and "Voila!" is meant for computers --- not human beings. "How does (did) one think of the steps that are to be carried out?" is relevant to the story. This need not be a factual/historical account as long as it is plausible and interesting!
As an example, consider the following (brief) account as how one can build up the notion of complex numbers from geometry.
In the study of the geometry of a (straight) line, one encounters numbers via translations to the left and right (traditionally, positive and negative respectively) which corresponds to addition of numbers. One also encounters scaling up or scaling down which corresponds to multiplication and division. What happens when one considers that plane?
One can certainly translate the plane in various directions. We can also rotate the plane and scale it. The recognition, that (after fixing a centre/origin and a unit "x" direction), each point in the plane can determine a rotation+scaling as well as a translation, allows us to define a multiplication as well as an addition of points on the plane. This operation creates the arithmetic operations making up the complex numbers.
One can also follow this up to see how regular polygons correspond to cyclotomic numbers and how constructible numbers involve solutions of quadratic equations. This can lead to a derivation of the recipe for the construction of a regular pentagon --- something which is rarely done in high school geometry books.
There are number of good books on popular mathematics:
- Mathematics for the Millions, by Lancelot Hobgen
- Figures for Fun, by Yakov Perelman
- Geometry and the Imagination, Hilbert and Cohn-Vossen.
- Mathematician's Delight, by W. W. Sawyer
- Godel, Escher, Bach, by D. Hofstader
There are a number of good books which are not popular mathematics:
- The Man Who Knew Infinity, by R. Kanigel
- The Men of Mathematics, by E. T. Bell
- The Code Book, by Simon Singh
- Fermat's Last Theorem, by Simon Singh
Too often (like in the current case!), one starts with the noble goal of writing popular mathematics, only to succumb to the (somewhat lazier) approach of writing about mathematical philosophy or mathematicians. This is not the only time I have done so! With faint heart, I recall an article that appeared in Science Age in 1984 on the work of Gerd Faltings --- an excellent article by Madhav Nori was vandalised by yours truly under pressure from the editors to make it "more accessible"!